ack1308,
The FE community has not decided on an accepted map with consistent distance measurements; to do this, they would need to agree on which distances to accept, which is a rather tedious process, as they expect a lot out of the measurements, which become loathsome to do on an individual level to the required degree of precision (they don't trust GPS which has already done the measurements).
To us Round Earth people, it's obvious why it's impossible to construct a flat map: enough distance measurements proves without a shadow of a doubt that the Earth is round. Therefore, it's fallacious to refer to the AE map and use it to disprove Flat Earth; many of them also admit that it's wrong and more work needs to be done to get a consistent map (of course, I've stated the RE opinion on the feasibility of consistency).
To address your acceleration point, the entire observable Universe accelerates with the Earth; however, the Earth shields us from the mysterious force that accelerates everything else, allowing us to be accelerated by the surface normal force of the Earth and for us to perceive weight and gravity. They invoke Einstein's equivalence principle (although they reject GR as a whole, it's not fallacious to use parts of RE theory) to show that no experiments can differentiate between a "downward" force on all of us and such acceleration. However, I prefer to interpret this invocation of Einstein's equivalence principle differently (I'll disclaim here that I know close to nothing about GR, besides some of the math used for it). Their model makes exactly the same predictions as one in which there is a magical force (violating the isotropy of space) pulling only those dwelling near a large, flat mass like the Earth "downward."
Now, the explanations tendered for the various effects explained by RE as the results of us sitting in a non-inertial reference frame (i.e. a spinning ball) are the variants of gravitation that FE accepts ("celestial gravitation"). Tom Bishop himself has stated that gravitation is what makes Foucault pendulums precess (it's on the wiki as well). Of course, if one does the mathematics on such things, one realizes that there actually need to be several, some arbitrarily applied, forces to correct for the vastly differing effects (distant stars on a pendulum, local variations in Earth's gravity); I do not know whether FE accepts this conclusion, and I don't claim to speak for them, but I do know that it is mathematically entailed from the assumption that "celestial gravitation" corrects for certain effects.
However, the most obvious failing of UA, to me, is the fact that the other planets are round. Again, I'm no authoritative expert on the FE hypothesis, but I believe they concede that their roundness is due to "celestial gravitation"; however, under UA + celestial gravitation, we'd actually see the other planets considerably flatter in the UA direction of acceleration, as the hypothesis is that the Earth's mass shields us from UA. Why shouldn't some of the mass of the other planets shield them as well? This would result in the downward (inward) gravitational force trying to pull everything together, but UA pushing on one axis in an imbalanced fashion. It's sort of like kicking a round football; while it is accelerating, you can see it flattens a bit. Of course, if FE acknowledges we can launch rockets to these planets, then a simple measurement of weight with the normal parallel to the direction of UA and another measurement with the normal orthogonal to the direction of UA will easily disprove the FE hypothesis. Then again, one could just turn the probe's camera toward Earth.
Of course, then there's always the Cavendish Experiment. FE can assert that gravity (an attraction between massive objects) is much weaker than we currently believe, but they have to give a reason why it deviates from the experimentally verified on a small scale F = (Gm1m2)/r^2 when the masses and radii get large. There is no satisfactory explanation. This experiment has been repeated enough that simply calling it "fake" is not enough.
I know nothing about the stars in the sky, so I won't address this point.